A New Strategy for the Lattice Evaluation of the Leading Order Hadronic Contribution to $(g-2{)}_{\mu }$

Abstract

A reliable evaluation of the integral giving the hadronic vacuum polarization contribution to the muon anomalous magnetic moment should be possible using a simple trapezoid-rule integration of lattice data for the subtracted electromagnetic current polarization function in the Euclidean momentum interval $Q^2>Q_{min}^2$, coupled with an $N$-parameter Padé or other representation of the polarization in the interval $0<Q^2<Q_{min}^2$, for sufficiently high $Q_{min}^2$ and sufficiently large $N$. Using a physically motivated model for the $I=1$ polarization, and the covariance matrix from a recent lattice simulation to generate associated fake "lattice data," we show that systematic errors associated with the choices of $Q_{min}^2$ and $N$ can be reduced to well below the 1% level for $Q_{min}^2$ as low as 0.1 GeV${}^2$ and rather small $N$. For such low $Q_{min}^2$, both an NNLO chiral representation with one additional NNNLO term and a low-order polynomial expansion employing a conformally transformed variable also provide representations sufficiently accurate to reach this precision for the low-$Q^2$ contribution. Combined with standard techniques for reducing other sources of error on the lattice determination, this hybrid strategy thus looks to provide a promising approach to reaching the goal of a sub-percent precision determination of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment on the lattice.

Comments: 27 pages, 9 figures